Chapter 6 Digital Filters CEN 352 Dr Nassim

Chapter 6 Digital Filters CEN 352, Dr. Nassim Ammour, King Saud University 1

Digital Filtering: Realization Definition: The Digital filter difference equation: Matlab Implementation: 3 -tap (2 ndorder) IIR filter CEN 352, Dr. Nassim Ammour, King Saud University 2

Example Given the DSP system with initial conditions and the input Solution: CEN 352, Dr. Nassim Ammour, King Saud University 3

Transfer Function Differential Equation: Z-Transform: Transfer Function: CEN 352, Dr. Nassim Ammour, King Saud University 4

Example: Transfer Function Given a DSP: Find the its transfer function H(z). Z-transform on both sides of the difference equation Z-Transform: factoring Y(z) on the left side and X(z) on the right side Rearrange: Transfer Function: Given H(z) : Find the difference equation of the system Rearrange: Differential Equation: Applying the inverse z-transform and using the shift property CEN 352, Dr. Nassim Ammour, King Saud University 5

Pole –Zero from Transfer Function • A digital transfer function can be written in the pole-zero form. • The z-plane pole-zero plot is used to investigate characteristics and the stability of the digital system. • Relationship of the sampled system in the Laplace domain and its digital system in the z-transform domain mapping: • The z-plane is divided into two parts by a unit circle. • Each pole is marked on z-plane using the cross symbol x, while each zero is plotted using the small circle symbol o. CEN 352, Dr. Nassim Ammour, King Saud University 6

Example: Pole-zero plot Given the digital transfer function: Plot poles and zeros The system is stable. The zeros do not affect system stability. CEN 352, Dr. Nassim Ammour, King Saud University 7

System Stability (Depends on poles’ location) • If the outmost poles of the DSP TF H(z) are inside the unit circle on the z-plane pole-zero plot, then the system is stable. • If the outmost poles are first-order poles of the DSP TF H(z) and on the unit circle on the z-plane pole-zero plot, then the system is marginally stable. CEN 352, Dr. Nassim Ammour, King Saud University 8

Example: System Stability Sketch the z-plane pole-zero plot and determine the stability for the system: Since the outermost pole is multiple order (2 nd order) at z = 1 and is on the unit circle, the system is unstable. CEN 352, Dr. Nassim Ammour, King Saud University 9

Digital Filter: Frequency Response Putting normalized digital frequency Magnitude frequency response CEN 352, Dr. Nassim Ammour, King Saud University Phase response 10

Frequency Response Example Problem Given the digital system with a sampling rate of 8, 000 Hz, determine the frequency response. Solution z-transform : (on both sides on the difference equation ) Frequency response: transfer function Magnitude frequency response Phase frequency response It is observed that when the frequency increases, the magnitude response decreases. The DSP system acts like a digital low-pass filter, and its phase response is linear. CEN 352, Dr. Nassim Ammour, King Saud University 11

Digital Filter: Frequency Response –contd. BASIC TYPES OF FILTERING Pass-band Ripple (frequency fluctuation) parameter Stop-band Ripple (frequency fluctuation) parameter Pass-band cut-off frequency Stop-band cut-off frequency High-pass filter (HPF) Low-pass filter (LPF) Matlab: Frequency Response: CEN 352, Dr. Nassim Ammour, King Saud University 12

Digital Filter: Frequency Response –contd. BASIC TYPES OF FILTERING Lower stop-band cut-off frequency Lower Pass-band cut-off frequency Higher pass-band Cut-off frequency Higher stop-band Cut-off frequency Band-pass filter (BPF) CEN 352, Dr. Nassim Ammour, King Saud University Band-stop filter (BSF) 13